Question

I am looking for a reference/idea, how this passage from Labesse's Snowbird Lecture "Introduction to endoscopy" pg.5 can be explained:

"We shall denote by $f_\pi$ a pseudo-coefficient for $\pi$, although it is highly non unique. But as regards invariant harmonic analysis this plays no role. In particular the orbital integrals are independent of the choice of the pseudocoefficient; they are also independent of the choice of the Haar measure on $G(F)$ but one has to use the canonical measure on the compact torus $G (F)$. The orbital integrals of $f_\pi$ are easily computed for regular semisimple: $$O_\gamma(f_\pi) =\begin{cases} \Theta_\pi(\gamma), & \gamma \mathrm{ elliptic}, \newline 0, &\mathrm{else},\end{cases} $$ where $\Theta_\pi(\gamma)$ is the character of $\pi$."

Here $G$ is a reductive group over a local field $F$ and $\pi$ a squareintegrable representation.

Answer 1

Existence of pseudo-coefficients for square-integrable representations (and the link with character values of the representations) is stated and proved in

D. Kazhdan, Cuspidal geometry of $p$-adic groups. J. Analyse Math. 47 (1986), 1–36.